"Till och med" is determining 'barnet', so these words is kept together. (Not just the mother and father, but 'even the child' is writing). But "Nu" is an adverb of time, which determines the whole sentence that follows: "Nu skriver barnet brev", "Nu" is emphasized in itself, therefore put at the start of the sentence, leading to the inversion.
I think in Swedish, it could either be the plural, or a sort of general usage, which Lundgren8 has written more about here: https://www.duolingo.com/comment/5824774
In English though, Even the child writes letter does not sound correct to me, which is probably why that translation isn't approved. What's your native language? Mine is obviously Swedish, but I think in English you'd either have to say a letter or letters (and a letter doesn't really fit the bill here either).
Late answer, but better late then never: We move "till och mer" in front of the noun we want to emphasize, so it is almost as in English: "Barnet skriver till och med brev". As you can see, in Swedish we have moved the verb out of the way, it has the standard second position.
I think of it as "up to and including ..."
You'll also see "från och med (date)" to mean "as of (date)" and in that context they're often abbreviated fr.o.m. and t.o.m.
The more colloquial meanings, e.g. when we're not talking about dates, are "even" and "furthermore."
Is there any way in Swedish to distinguish between "Even the child is writing letters," e.g. it's a letter writing party and everyone is writing letters right now, even the child; and "Even the child writes letters," e.g. they're a family of avid letter-writers who all send each other a letter every day, even the child?
Is "Till och med barnet skriver brev" closer to one of those translations than the other?
As I understand your question, it is not about 'Even the child' (which is the same in the two sentences) - but about putting the verb in a continuous tense or not. We don't have the English -ing-continuous. The present tens contains both. BUT - we have other ways of conveying what is going on. E.g. Even the child is writing letters = Till och med barnet håller på och skriver brev/bokstäver.
Yes, sorry for the confusion. It's mildly frustrating to me that Duolingo's English translations almost always use the present tense when the present progressive/continuous would make more sense. It makes it difficult to visualize and therefore understand the meaning of the Swedish sentence. Then here we see a sentence where the present tense makes sense... and Duolingo uses the present progressive/continuous instead.